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Is there any reference about solvability theory of quasilinear elliptic equations on closed manifolds?

In particular, I am looking for solvability condition for function $f$ of following equation $\Delta u+ c |\nabla u|^2=f$ on a closed manifold $M$, $c$ is a positive constant. As far as I understand $\int_M\, \Delta u+c |\nabla u|^2\, dV=\int_M\, c |\nabla u|^2\, dV\geq 0$, so at least $\int_M\, f\, dV\geq 0$, is that enough? Is there any sufficient condition for solvability?

Thanks a lot.


Now seems like the statement of the problem can be simplified as following

Under what condition for $f$, there exists positive solution $u$ for the Schrodinger equation $\Delta u+f u=0$ on a closed manifold.

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    $\begingroup$ At the maximum value of $u$, $f$ must be negative, so you would also need an $f$ that changes sign -- i.e., the integral condition is not sufficient. $\endgroup$ – Rbega Oct 1 '14 at 20:00
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No. Consider the case of periodic solutions in one dimension, and let c=1. That is, we have the equation $$u''+(u')^2=f.$$ Multiply both sides by $e^u$. You end up with $$(e^u)''=e^uf.$$ Now if, for instance, $f$ is positive, then the right hand side is positive, and the left hand side has zero average, which cannot be.

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  • $\begingroup$ Thanks @Michael, then is there any sufficient condition? $\endgroup$ – Joe Oct 1 '14 at 20:17
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Let $w=e^{cu}$, then the equation for $w$ is $$\Delta w-cfw=0,$$ which is a Schrodinger equation. A necessary for a Schrodinger equation to have a positive solution is Barta's lemma: $$\lambda_1(-\Delta+cf)\geq 0.$$

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  • $\begingroup$ Is it sufficient? $\endgroup$ – Joe Oct 1 '14 at 22:12
  • $\begingroup$ Yes, I think it is necessary and sufficient. You can check Barta's Lemma in this paper (Lemma 3.1): arxiv.org/abs/1211.2705 $\endgroup$ – littlelittlelittle Oct 1 '14 at 22:20

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